High-throughput RNA-sequencing (RNA-seq) technology provides an attractive platform for gene expression analysis. In many experimental settings, RNA-seq read counts are measured from matched samples or taken from the same subject under multiple treatment conditions. The induced correlation therefore should be evaluated and taken into account in deriving tests of differential expression. We proposed a novel method 'PLNseq', which uses a multivariate Poisson lognormal distribution to model matched read count data. The correlation is directly modeled through Gaussian random effects, and inferences are made by likelihood methods. A three-stage numerical algorithm is developed to estimate unknown parameters and conduct differential expression analysis. Results using simulated data demonstrate that our method performs reasonably well in terms of parameter estimation, DE analysis power, and robustness. PLNseq also has better control of FDRs than the benchmarks edgeR and DESeq2 in the situations where the correlation is different across the genes but can still be accurately estimated. Furthermore, direct evaluation of correlation through PLNseq enables us to develop a new and more powerful test for DE analysis. Application to a lung cancer study is provided to illustrate the practical utilities of our method. An R package implementing the method is also publicly available.